Construction of Virtual Neuron and Consolidation of Sleep and Memory Process– A Molecular Docking and Biomathematical Approach

The relationship between sleep and memory never fails to fascinate human beings as we always keep curious about how the memory form and consolidate [1]. Nevertheless, recent developments in molecular genetics, neurophysiology, and the cognitive neurosciences have produced a striking body of research that provides converging evidence for an important role of sleep in learning and the reprocessing of memories. Also, sleep deprivation has become a prevalent public health epidemic with a wide range of harmful consequences, including memory and cognitive impairment [2], some scientists tend to focus attention on the neurobiological root of this universal human experience.

Effects of sleep stage on consolidation early study of different sleep stages in memory consolidation in rats mainly focused on REM sleep (Random Eye Movement sleep) and the consequences of REMD (REM sleep deprivation) by repeatedly waking subjects at the first signs of REM sleep. All these memory tasks were typically emotionally loaded as it has been proved that REM sleep priority to improve the consolidation of memory emotions [3,4], and it turned out that REMD is only valid for a specific period of time after learning — the so-called “REM sleep window” [5,6].

The first evidence of the causal role of SWS (Slow Wave Sleep) reactivation during memory consolidation comes from the study of human spatial location in the presence of odor [7]. Reactivation activates hippocampal memory redistribution to new cortical storage locations. On the other hand, another hypothesis, named ‘sequential hypothesis’, which argues that the best benefit of sleep for consolidating declarative and non-declarative memory occurs when SWS and REM sleep occur one after the other is also verified by the cross effect of SWS/REM on declarative/procedural memory consolidation [8,9].

Sleep’s promotion on memory consolidation

Sleep is characterized by the rapid occurrence of REM sleep and non-REM sleep, including slow wave sleep (SWS, Phases 3 and 4) and light sleep Phases 1 and 2 (see Figure 1, part a). In humans, the first part of the night (early sleep) is characterized by a high amount of SWS, while REM sleep is dominant in the lower half (late sleep). SWS and REM sleep are characterized by a specific pattern of electric field potential oscillations (see Figure 1, part b) (Figure 1).

In subsequent rapid eye movement (REM) sleep, the brain system operates in a “separated” mode, which is also associated with separation between long-term and temporary storage. This enables a synaptic consolidation of the local encapsulation process, which reinforces the memory characterization of system integration during previous SWS (thick line). In general, memory gets the best results from slow wave sleep and rapid eye movement sleep. However, due to their different characteristics, integrated declarative memory (which combines the characteristics of different memories in different memory systems) theoretically benefits more from SWS-related system integration, with specific and discrete procedural memory benefits more from REM sleep — associated synapse consolidation in the local brain circuit [10] (Figure 2).

There are currently two hypotheses that explain the mechanism of memory consolidation during sleep. Synaptic homeostasis hypothesis [11] is a by-product of the overall synaptic reduction that occurs during sleep. The active system consolidation hypothesis suggests that the active consolidation process is caused by selective reactivation of memory in sleep [12]. The former use is to serve a global downscale of synaptic strength by the slow oscillations to achieve sustainable levels in energy and tissue volume requirements and allow for the repetitive use of synapses in future coding, and the process of the latter can be depicted [13,14]. Therefore, we have the two-stage model and sequential contributions of SWS and REM sleep on memory consolidation (Figure 3).

Memory impairment after sleep deprivation

Marion and his colleagues once used electrophysiological, molecular and behavioural indices to non-invasively study LTP-like plasticity in humans after sleep and sleep deprivation. The outcome shows that LTP-like plasticity decreases after sleep deprivation verified by declined MEP amplitudes, lower potential modulator level and less correctly recalled word-pairs during the declarative memory task [15]. For further studying, Tudor et al extended that 5 hours of sleep loss attenuated both mTORC1-mediated phosphorylation of 4EBP2 and the interaction between eukaryotic initiation factor 4E (eIF4E) and eIF4G in the hippocampi of sleepdeprived mice using an in vivo protein translation assay. What is more, by increasing the abundance of 4EBP2 in hippocampal excitatory neurons before sleep deprivation the abundance of phosphorylated 4EBP2 increased, and the amount of eIF4EeIF4G interaction restored and hippocampal protein synthesis to that seen in mice that were not sleep-deprived and prevented the hippocampus-dependent memory deficits associated with sleep loss [16]. Effects of sleep deprivation on protein synthesis procedure is also verified [17] (Figure 4).

And the molecular neurobiology mechanism of sleep on protein synthesis procedure is also verified [18]. Next, we consider the neuronal functions in sleep and memory consolidation process by the following fractional-order differential equations:

where 0

Suppose that some ractional-order differential equations are response systems of the neuronal functions in sleep and memory consolidation process:

where are the estimated values of the parameters a, b, c, h, r on the system (3.1) respectively; and 0<β_i <1,(i= 1,2,3, 4) is a parameter describing the order of the system (3.2); are the sleeping time in the Caputo sense, and are the status vectors of system (3.1) and system (3.2) respectively. is the controller.

If α_i = 0.98, (i =1, 2,3, 4), a = 35, b = 3, c= 12, h= 7, r= 0.5, then the diagram of the attractors of system (3.1) can be seen in (Figure 5).

Neurotransmitters signals Transportation

Neurotransmitters are chemical agents secreted at the end of axons of nerve cells that diffuse across the synaptic gap and transmit information to adjoining cells such as neurons, muscle cells, and glands, by altering their electrical state or activity. There are many neurotransmitters with a variety of structures and functions.

In order to identify the neuronal functions in sleep and memory consolidation process shown above, the sleeping time step is taken as h= 0.0025s, and

The time taken for the simulation, and the order parameters, with the initial state value of system (3.1) is taking system, to select the system x (0) = (2, 0,1,1), the initial state value of system (3.2) is y(0)=(-4,-2,1,-5). Therefore, the sleeping process between system (3.1) and system (3.2) is shown in the following (Figure 6).

According to the definition of neurotransmitters signals transportation, suppose that the neurotransmitters signals transportation is e= x+ y. If for any x (0), y(0) satisfy the condition then we say that system (3.1) and system (3.2) achieve modifiable structures.

On the basis of adaptive control methods, we can give the virtual neurotransmitter by serotonin and acetylcholine systems in central nervous system involving sleep and memory:

where 4). If and system (3.1) and system (3.2) achieve modifiable structures indicating memory process.

If we put (4.1) and system (3.1) to system (3.2), then the following error equations can be obtained between the groups for some fractional differential equations:

where are the parameter estimation errors.

Next, according to (4.2), we design the adaptive update law for each parameter estimation error:

where 0

According to and (4.3), we can get the parameters of the adaptive control law:

According to (4.2) and (4.3), we get the total error of the system:

Where

Then we consider Eq. (4.5), and expand the formula, we obtain:

Setting P = E9. Then we obtain the following result:

AP=PA^T

=A+A^T= -Q

where ki > 0, (i=1,2,3,4) Q=diag 2k₁, 2k₂, 2k₃, 2k₄, 0, 0, 0, 0, 0).

It is easy to see that Q=diag 2k₁, 2k₂, 2k₃, 2k₄, 0, 0, 0, 0, 0 is a semi-positive definite matrix. Then, the state variable of (4.5)

is asymptotically stable, that is,approach zero asymptotically with time.

Therefore, we achieve the neuronal functions in sleep and memory consolidation process by a number of adaptive robust set of fractional differential equations anti-synchronization indicating memory process.

Chemical and Physicael Propertis

In this part, mainly through the mathematical method, the preliminary screening of the data in Table 1 for the chemical and physicael propertis of neuron, and respectively by MATLAB and SPSS 22 software to the data provided by principal component analysis, that can only find that the influence factors on the quality of sleep and memory process standard, but can’t find out what factors. Then we use the binary logistic regression model to process the data (Table 1&2).

First, we extract the data from the above Table 2 to make the 12*7 Table, and then use the principal component analysis method in MATLAB software (The code is shown in the last Annex 1 of this article). We find out that the most influential factors are Formal Charge, Heavy Atom Count, Defined Atom Stereocenter Count. Finally, we analyze the data in Table 2, draw the broken line diagram (The code is shown in the last Annex 2 of this article), which is the following Figure 7, and we find out that the fifth neurotransmitters GABA is the most prominent one [19-62] (Figure 7).

Table 1:Chemical and physical properties of the neuron.

Table 2: The compression of the chemical and physicael propertis of the neuron.

For X ≠ 0 and natural number n, taking the logarithmically differential into consideration yields

where and ψ (x + y) respectively.

Furthermore, differentiating directly gives

Making use of (2.11) and (2.13) shows that for all n∈N and any fixed y >0, the double inequality

holds for all x∈(−y,∞) \{0} and α ∈(−∞,∞) For any fixed y∈(0,∞) , let u(t) and v(t) be defined on (−∞,∞) by

respectively.

Differentiating u(t) and v(t) directly, we obtain

Therefore, for given y∈(−∞,∞) , we have

From (5.6) and (5.7), we conclude that for all t > 0 we obtain u (t)> 0, and

From (5.3) and (5.8) -(5.9), it is easy to see that

For all n∈N x∈(−y,∞) \ {0}

On the one hand, if x∈(0,∞) then the inequalities (5.10) can be equivalently changed into

From (5.1), then simple computation shows that

and

for all k ∈N and all x > 0. Therefore, (5.14) and (5.15) imply

for all n∈N and all x > 0.

Hence, if either for given 0 < y <1 or α ≥1 for given y ≥1, the function (1.13) is strictly logarithmically completely monotonic with respect to x on

for n∈N

In view of (5.13), we can conclude that

for n∈N It is obvious that (5.18) is equivalent to that (5.14) and (5.15) hold for any given y> 0 and x∈(−y,0) .Therefore, it is easy to prove similarly that (5.16) is also valid on x∈(−y,0) for any given y>0 and all n∈N As a result, we can conclude that nervousness has the strongest influence on sleep and memory quality.

SWS and REM sleep complement each other to optimize memory integration. In the SWS process in which slow oscillations induce a broad synchronization of neuronal activity, active system consolidation combines newly encoded memory with preexisting long-term memory to induce conformational changes in their respective expressions. System integration (priority affects explicit coding of behavior-related information) synergizes with global synaptic downscaling, primarily to rule out saturation of synaptic networks. Subsequent REM sleep, characterized by desynchronization of the neural network, may reflect the detachment of the memory system—possibly stabilizing the transformed memory by consolidating undisturbed synapses.

Sleep deprivation suppressed activation of the kinase complex mTORC1 which leads to a decrease in phosphorylation of 4EBP2, leading to disruption of protein translation, reduced neuroplasticity, and ultimately cognitive impairment. Restoring protein synthesis by increasing the amount of phosphorylated 4EBP2 protein in the hippocampus, a function normally performed by mTORC1 can compensate the memory impairment caused by sleep deprivation. Also, some alternative mechanisms may actuate as well in the protein synthesis reduction which needs a further studying.

The development of neurotransmitters and its complex functions during sleep and memory process are influenced by numerous factors. In this study, some mathematical speculations have been proposed on the basis of structural and functional characteristics of virtual neuron (especially the physiological phenomena of human beings) with a molecular docking and biomathematical approach to formulate some speculations to the consolidation of sleep and memory process. This could pave a way to formulate more mathematical speculations related with neuron, and finally these data and approaches will be useful for constructing virtual neuron with the help of biomathematics.

We have no conflict of interests to disclose and the manuscript has been read and approved by all named authors.

We would like to express my gratitude to all those who helped us during the writing of this article.

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